Why fit a surface
Each option's implied volatility comes from its own quote, and quotes are noisy: wide bid/ask spreads, stale prices and thinly traded strikes all add error. Two strikes a dollar apart can show volatilities that clearly do not belong on one smooth curve.
A volatility surface is a smooth function of strike and expiration fitted through those points. Skew, term structure and smile are all slices of it.
SVI
SVI (stochastic volatility inspired) is a widely used formula for fitting one expiration's smile with five parameters. Gatheral and Jacquier (2014) set out conditions under which an SVI surface is free of static arbitrage, meaning its prices do not allow a riskless profit.
Two checks matter. The butterfly check makes sure the fitted smile implies sensible, non-negative probabilities across strikes. The calendar check makes sure total variance (volatility squared times time) does not fall as expiration lengthens. Example: an annualized volatility of 30% at one month and 25% at three months passes, because total variance rises from 0.0075 to about 0.0156. An inverted term structure is not by itself an arbitrage.
Reading the outputs
A fitted surface lets you read a smoothed implied volatility at any strike, compare wings with the middle, and compare puts and calls at matching deltas. A variance-based volatility, calculated from a whole strip of options, is a different number from at-the-money implied volatility.
Caveat: a smooth fit can still be wrong where quotes are sparse, especially in the far wings. A failed check can mean bad quotes, a poor fit, or extrapolation beyond the data, so it is a reason to investigate rather than an automatic verdict on the data.